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I strangely could not find a reference for this. What are some (if any) model structures on the category of small $A_{\infty}$ categories, with weak equivalences quasi-equivalences. Same question in the case underlying chain complexes are unbounded, ungraded.

Related question, let $M$ denote the category of small $A_\infty$ categories, $M^D$ category of $D$ shaped diagrams with $D$ Reedy with fibrant constants. Think of $M^D$ as a homotopical category with weak equivalences objectwise quasi-equivalences, similarly with $M$. Is the functor $$colim M^D \to M$$ homotopical? It is not homotopical in the dg world, but it seems it might be in the $A_\infty$ world. If it is not homotopical and the answer to question one is "none" does the derived functor $hocolim: ho M^D \to ho M$ exist?

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Here's a guess: Take weak equivalences as stated, and have fibrations be functors surjective (degreewise) on Hom complexes and induce an isofibration on homotopy categories (i.e. RLP with respect to the inclusion 0 --> J where J is the groupoid with two objects 0, and 1 with a unique isomorphism between them). This gives the correct model structure on dg-categories over a ring, for example... –  Dylan Wilson Apr 14 '13 at 0:31
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Another idea is to ramp up Lurie's nerve construction for dg-categories to work for A_infty categories (he says this can be done), and then maybe transfer the model structure from the Joyal model structure? Dunno if this would work though... –  Dylan Wilson Apr 14 '13 at 0:32
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up vote 9 down vote accepted

Look at Lefevre-Hasegawa PhD Thesis (unfortunately in French) at http://arxiv.org/abs/math/0310337, where this is done. For a reference in English, you can look there: http://math.unice.fr/~brunov/HomotopyTheory.pdf , where you have to restrict to the the Koszul operad $As$ of associative algebras and then consider "mutliple objects", ie $A_\infty$-categories.

Strictly speaking, you cannot have a model category structure on $A_\infty$-categories since not all the equalizers exist. But "all" the other axioms hold. In this case, weak equivalences are $\infty$-quasi-isomorphisms (does this correspond to your quasi-equivalences?), cofibrations are $\infty$-monomorphisms (the first component map is a monomorphism) and fibrations are $\infty$-epimorphisms (the first component map is an epimorphism). In this model category structure, everybody is fibrant and cofibrant. (I have an extended version of loc. cit. including this, that I can share.)

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This is great! Glad to hear about this work –  David White Apr 14 '13 at 14:20
    
Thanks. This is close to what I was expecting, except for quasi-isomorphism vs quasi-equivalence. Which is a little strange since geometrically (in the setup of Fukaya categories) quasi-equivalences are a lot more natural than quasi-isomorphisms. Not having all equalizers does not seem to affect the second half of question, and it seems that the colim functor is indeed homotopical provided we replace quasi-equivalence by quasi-isomorphism in the question. Am I right? –  yasha Apr 14 '13 at 15:37
    
Why unfortunately? –  Fernando Muro Apr 14 '13 at 15:53
    
Well "unfortunately" because yasha might not read French but still would like to understand the result. That might take long if he first learn French (which he should fo) and then begin reading Lefevre-Hasegawa thesis. :) –  Bruno V. Apr 14 '13 at 22:19
    
I don't quite agree with you, Bruno. I like that French has survived for mainstream science, unlike the rest of languages except for English. –  Fernando Muro Apr 16 '13 at 18:37
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